Finite Group Databasebeta

← #10826322

Subgroups — #10826322

Order 768.

1760 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#548C211yesmaximal, characteristic
2C23×#216C211yesmaximal, characteristic, complemented
3C2×#5239C212yesmaximal, complemented
4C2×#5040C211yesmaximal, characteristic, complemented
5#82349C211yesmaximal, characteristic, complemented
6#72132C211yesmaximal, characteristic, complemented
7#71729C212yesmaximal, complemented
8#71700C212yesmaximal, complemented
9#71681C211yesmaximal, characteristic, complemented
10#71632C212yesmaximal, complemented
11#71610C211yesmaximal, characteristic, complemented
12#23859—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C23×Dic6C2211yescharacteristic
16C3×#548—24no—
17C3×#548—24no—
18C2×C3×#124—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×#121—21no—
23C2×C3×#121C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2211yescharacteristic
27C22×#216—24no—
28C22×#216—28no—
29C22×#216C2212yes—
30C22×#216C2212yes—
31C22×#214C2212yes—
32C2×#924C2212yes—
33C23×C3⋊C8C2211yescharacteristic
34#5239—28no—
35C2×#835C2212yes—
36C2×#835C2212yes—
37#5220—21no—
38C2×#832C2212yes—
39#5212C2211yescharacteristic
40C2×#830C2212yes—
41C2×#830C2211yescharacteristic
42#5040—24no—
43#4979—21no—
44C2×#810C2212yes—
45#4947—21no—
46#4946—21no—
47#4929C2212yes—
48#4529C2212yes—
49#4524C2212yes—
50C23×Q16—31no—
51C2×#548D311yescharacteristic
52C2×#452—32no—
53C2×#448—31no—
54#2409—31no—
55#2015—32no—
56#2001—32no—
57#1893—31no—
58#1878—32no—
59#1869—31no—
60#1787—31no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—24no—
64C2×C3×#41—42no—
65C22×C3×Q8—21no—
66C22×C3×Q8—22no—
67C22×C3×Q8—24no—
68C22×C3×Q8—42no—
69C22×C3×Q8D414yes—
70C22×C3×Q8C2311yescharacteristic
71C22×C3×Q8C2311yescharacteristic
72C22×C3×D4—21no—
73C22×C3×D4—21no—
74C22×C3×D4D411yescharacteristic
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C23×Dic3—21no—
81C22×Dic6—22no—
82C22×Dic6—24no—
83C22×Dic6—22no—
84C22×Dic6—21no—
85C22×Dic6—42no—
86C22×Dic6C2312yes—
87C3×#124—28no—
88C3×#124—22no—
89C3×#124—22no—
90C3×C22⋊Q8—28no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×#121—28no—
94C3×#121—22no—
95C3×#121D414yes—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×D4—28no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4D411yescharacteristic
110C2×C3×C22⋊C4D411yescharacteristic
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C2×#217—21no—
114C2×#217—22no—
115C2×#216—232no—
116C2×#216—24no—
117C2×#216—24no—
118C2×#216—24no—
119C2×#216—24no—
120C2×#216—24no—
121C2×#216D418yescomplemented
122C2×#214—24no—
123C2×#214—24no—
124C2×#214—42no—
125C2×C4⋊Dic3C2312yes—
126#924—24no—
127#924—42no—
128C2×Dic3⋊C4—22no—
129C22×C3⋊C8—22no—
130C22×C3⋊C8—24no—
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2312yes—
133#835—24no—
134#835—42no—
135#832—24no—
136C3⋊#95—22no—
137C3⋊#95—22no—
138C3⋊#95—21no—
139#830—22no—
140#830—24no—
141#810—24no—
142C22×#41C2×D311yescharacteristic
143C23×Q8—31no—
144C23×Q8C2×D311yescharacteristic
145C22×Q16—32no—
146C22×Q16—32no—
147C22×Q16—68no—
148C22×Q16—64no—
149C22×SD16—32no—
150C23×C8—31no—
151#548—24no—
152#548—24no—
153C2×#124—21no—
154C2×#124C2×D311yescharacteristic
155C2×C22⋊Q8—21no—
156C2×C22⋊Q8—32no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×#121—21no—
159C2×#121C2×D311yescharacteristic
160C2×C4×Q8C2×D311yescharacteristic
161C2×C4×D4—21no—
162C2×C4×D4C2×D311yescharacteristic
163#452—68no—
164#448—64no—
165C2×#103—32no—
166C2×#103—32no—
167C2×#103—32no—
168C2×#102—32no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—32no—
171#395—61no—
172#393—61no—
173#391—32no—
174#386—61no—
175#385—61no—
176#380—31no—
177#340—32no—
178#336—32no—
179C24×C3—21no—
180C24×C3C2×D411yescharacteristic
181C24×C3C2×D411yescharacteristic
182C3×#41—48no—
183C3×#41—44no—
184C3×#41—44no—
185C2×C3×Q8—24no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—28no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—44no—
193C2×C3×Q8—44no—
194C2×C3×Q8—48no—
195C2×C3×Q8C2×D414yes—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8C2×D412yes—
198C2×C3×D4—24no—
199C2×C3×D4—22no—
200C2×C3×D4—24no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—21no—
205C2×C3×D4—22no—
206C2×C3×D4—44no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4C2×D412yes—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—44no—
222C22×C3×C4—44no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—22no—
231C22×Dic3—22no—
232C22×Dic3—44no—
233C22×Dic3—42no—
234C2×Dic6—28no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—22no—
238C2×Dic6—22no—
239C2×Dic6—22no—
240C2×Dic6—48no—
241C2×Dic6—44no—
242C2×Dic6—44no—
243C2×Dic6—42no—
244C2×Dic6C2×D414yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D412yes—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—22no—
256C3×C42—24no—
257C3×C42—21no—
258C3×C42—21no—
259#217—44no—
260#217—42no—
261#216—216no—
262#216—216no—
263#214—48no—
264C4⋊Dic3—22no—
265C4⋊Dic3—22no—
266Dic3⋊C4—44no—
267C2×C3⋊C8—28no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×C3⋊C8C2×D414yes—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—24no—
277C2×#41—42no—
278C22×Q8—21no—
279C22×Q8—22no—
280C22×Q8—24no—
281C22×Q8—32no—
282C22×Q8—42no—
283C22×Q8—64no—
284C22×Q8—61no—
285C22×Q8—62no—
286C22×Q8—62no—
287C22×Q8—122no—
288C22×Q8#6714yes—
289C22×Q8C22×D311yescharacteristic
290C22×Q8C22×D311yescharacteristic
291C22×D4—21no—
292C22×D4—21no—
293C22×D4#6711yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299C23×C4C22×D311yescharacteristic
300C2×Q16—38no—
301C2×Q16—64no—
302C2×Q16—64no—
303C2×Q16—632no—
304C2×Q16—64no—
305C2×Q16—64no—
306C2×Q16—64no—
307C2×SD16—64no—
308C2×SD16—64no—
309C2×SD16—122no—
310C22×C8—32no—
311C22×C8—32no—
312C22×C8—64no—
313C22×C8—62no—
314#124—28no—
315#124—22no—
316#124—22no—
317C22⋊Q8—28no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—64no—
321C22⋊Q8—122no—
322#121—28no—
323#121—22no—
324#121#6714yes—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×D4—28no—
328C4×D4—22no—
329C4×D4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—62no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4#6711yescharacteristic
343C2×C22⋊C4#6711yescharacteristic
344C2×C42—21no—
345C2×C42C22×D311yescharacteristic
346#103—64no—
347#103—64no—
348#103—122no—
349#102—64no—
350C22⋊C8—64no—
351C22⋊C8—62no—
352#95—61no—
353#95—62no—
354#95—62no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—22no—
363C23×C3—42no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—42no—
367C23×C3#12012yes—
368C23×C3#13512yes—
369C23×C3#13712yes—
370C23×C3C22×D411yescharacteristic
371C3×Q8—22no—
372C3×Q8—24no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—48no—
376C3×Q8—42no—
377C3×Q8#12014yes—
378C3×D4—24no—
379C3×D4—48no—
380C3×D4—44no—
381C3×D4—48no—
382C3×D4—41no—
383C3×D4—41no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—48no—
400C2×C3×C4—48no—
401C2×C3×C4—42no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—42no—
407C2×C3×C4#12012yes—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4H(5,𝔽2)12yes—
412C2×Dic3—24no—
413C2×Dic3—48no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416C2×Dic3—42no—
417C2×Dic3—42no—
418C2×Dic3—42no—
419Dic6—24no—
420Dic6—24no—
421Dic6—48no—
422Dic6—42no—
423Dic6—42no—
424C3⋊C8—24no—
425C3⋊C8—42no—
426C24—21no—
427C24C2×#6711yescharacteristic
428C24C2×#6711yescharacteristic
429#41—48no—
430#41—44no—
431#41—44no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—28no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—34no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—68no—
444C2×Q8—62no—
445C2×Q8—62no—
446C2×Q8—62no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—124no—
450C2×Q8—128no—
451C2×Q8—124no—
452C2×Q8—122no—
453C2×Q8D3×D414yes—
454C2×Q8C2×#6712yes—
455C2×Q8C2×#6712yes—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×#6712yes—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—124no—
484C22×C4—122no—
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6712yes—
490C22×C4C2×#6712yes—
491C22×C4C23×D311yescharacteristic
492Q16—616no—
493Q16—616no—
494SD16—128no—
495C2×C8—34no—
496C2×C8—68no—
497C2×C8—62no—
498C2×C8—62no—
499C2×C8—62no—
500C2×C8—62no—
501C2×C8—62no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—62no—
507C4⋊C4—62no—
508C4⋊C4—124no—
509C4⋊C4C2×#6712yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—124no—
517C22⋊C4—122no—
518C42—24no—
519C42—21no—
520C42—21no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—24no—
524C22×C3—24no—
525C22×C3—42no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—82no—
534C22×C3—82no—
535C22×C3#45214yescomplemented
536C22×C3C2×#12011yescharacteristic
537C22×C3D4212yes—
538C22×C3C2×#13511yescharacteristic
539C22×C3C2×#13711yescharacteristic
540C22×C3#57812yes—
541C3×C4—22no—
542C3×C4—24no—
543C3×C4—41no—
544C3×C4—42no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—41no—
548C3×C4C2×#12011yescharacteristic
549C3×C4#53411yescharacteristic
550C3×C4D4212yes—
551Dic3—44no—
552Dic3—82no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—21no—
560C23—22no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23#95012yes—
566C23#95112yes—
567C23#95312yes—
568C23C22×#6711yescharacteristic
569Q8—24no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—42no—
575Q8—64no—
576Q8—64no—
577Q8—128no—
578Q8—122no—
579Q8—122no—
580Q8#93714yes—
581D4—24no—
582D4—48no—
583D4—44no—
584D4—48no—
585D4—41no—
586D4—41no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—64no—
611C2×C4—128no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4#95312yes—
618C2×C4C2×D3×D412yes—
619C2×C4#100412yes—
620C2×C4C22×#6711yescharacteristic
621C2×C4C22×#6711yescharacteristic
622C8—64no—
623C8—122no—
624C2×C3—24no—
625C2×C3—81no—
626C2×C3—82no—
627C2×C3#240311yescharacteristic
628C2×C3C2×#45211yescharacteristic
629C2×C3#301111yescharacteristic
630C2×C3#327012yes—
631C2×C3#328512yes—
632C22—22no—
633C22—24no—
634C22—24no—
635C22—22no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22#523914yescomplemented
647C22D4×#6712yes—
648C22C2×#95011yescharacteristic
649C22C2×#95111yescharacteristic
650C22#583812yes—
651C22C2×#95311yescharacteristic
652C4—22no—
653C4—24no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—124no—
660C4—242no—
661C4C2×#93711yescharacteristic
662C4D4×#6712yes—
663C4#580411yescharacteristic
664C3#2385911yescharacteristic, complemented, Sylow
665C2—24no—
666C2—81no—
667C2—82no—
668C2#8354612yes—
669C2#8354912yes—
670C2#8405211yescharacteristic
671C2C2×#523911yescharacteristic
672C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#548C2
socleC23×C3C22×D4
hypercenterC2×#548D3
hypocenterC3#23859
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#216C2semidirect
3. C2×#5239C2semidirect
4. C2×#5040C2semidirect
5. #82349C2semidirect
6. #72132C2semidirect
7. #71729C2semidirect
8. #71700C2semidirect
9. #71681C2semidirect
10. #71632C2semidirect
11. #71610C2semidirect
121. C2×#216D4semidirect
535. C22×C3#452semidirect
646. C22#5239semidirect
664. C3#23859semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5482
C23×#2162
C2×#52392
C2×#50402
#823492
#721322
#717292
#717002
#716812
#716322
#716102
#238593